ZpZ

ZpZ is a nonstandard or informal notation that appears in some mathematical texts to denote the quotient ring Z/pZ, the ring of integers modulo p. The parameter p is a positive integer, commonly a prime, and the elements of Z/pZ are the residue classes [0], [1], ..., [p−1], with addition and multiplication performed modulo p.

If p is prime, Z/pZ is a field, isomorphic to the finite field GF(p). If p is composite, the structure is a finite ring with zero divisors. The notation emphasizes the modular arithmetic aspect: computations are carried out by reducing results modulo p.

Notation caveats: Z_p is widely used to denote the ring of p-adic integers, which is a very different object from Z/pZ. Consequently, ZpZ can be ambiguous. Most authors use Z/nZ or Z/pZ for the quotient ring, and reserve F_p or GF(p) for the field of p elements when p is prime.

Common alternatives and related concepts include Z/nZ, Z_n, and F_p, depending on context and audience. In applications, Z/pZ and related finite fields are foundational in modular arithmetic, number theory, cryptography, and coding theory, where working modulo p yields useful algebraic and computational properties.